Generating Functions

We investigate the probability that a random composition (ordered partition) of the positive integer $n$ has no parts occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{`forbidden set'}$ $A$ of multiplicities. This probability is also studied in the related case of samples $\Gamma =(\Gamma_1,\Gamma_2,\ldots, \Gamma_n)$ of independent, identically distributed random variables with a geometric distribution. Nous examinons la probabilité qu'une composition faite au hasard (une partition ordonnée) du nombre entier positif $n$ n'a pas de parties qui arrivent exactement $j$ fois, où $j$ appartient à une série interdite, finie et spécifiée $A$ de multiplicités. Cette probabilité est aussi étudiée dans le cas des suites $\Gamma =(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ de variables aléatoires identiquement distribuées et indépendantes avec une distribution géométrique.

The neighbourhood prime labelling of a graph G is defined as a function f : V (G) −→ {1, 2, 3, ..., n} which is bijective and if for every vertex of G with degree greater than 1, gcd {f(u) : u ∈ N(v)} = 1. A graph is called neighbourhood prime if it admits neighbourhood prime labelling. In this paper we introduce m-corona and prove that the corona products Pn Pn, the shadow graph of paths Pn, m-corona of paths Pn are all neighbourhood prime graphs. AMS Subject Classification: 05C78

In 1862, Wolstenholme [16] proved that the above congruence holds modulo p for any prime p ≥ 5, which is known as the famous Wolstenholme’s theorem. It is well-known that Wolstenholme’s theorem is a fundamental congruence in combinatorial number theory. We refer to [12] for various extensions of Wolstenholme’s theorem. In the past few years, (q-)congruences for sums of binomial coefficients have attracted the attention of many researchers (see, for instance, [2, 3, 6–11, 14, 15]). In 2011, Sun and Tauraso [15] proved that for any prime p ≥ 5,

We analyze the problem of finding combinatorial interpretations of holonomic recurrences and show some techniques that were used to achieve known results. We also try to find a combinatorial interpretation of the holonomic recurrence for trees of lambda terms in the de Bruijn notation.

In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution.

Laguerrian derivatives and related autofunctions are presented that allow building new special functions determined by the action of a differential isomorphism within the space of analytical functions. Such isomorphism can be iterated every time, so that the resulting construction can be re-submitted endlessly in a cyclic way. Some applications of this theory are made in the field of population dynamics and in the solution of Cauchy’s problems for particular linear dynamical systems.